# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 44 - 51 of 101

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## Question number: 44

Appeared in Year: 2016

### Describe in Detail

Let be a fined positive integer. Let

Evaluate

### Explanation

Consider the expansion ………. . eq. (1)

Integrating both sides of (1) within limits , we get

… (114 more words) …

## Question number: 45

Appeared in Year: 2007

### Write in Short

Let be a seventh root of unity. Write down a polynomial equation of degree satisfied by w.

## Question number: 46

» Integration in Complex Plane

Appeared in Year: 2010

### Describe in Detail

Let C be the semicircle where varies from , in the complex plane. Evaluate:

### Explanation

Equation of given circle is

## Question number: 47

» Residues of Complex Function

Appeared in Year: 2009

### Write in Short

Let

Write down the residues of at each of its poles

## Question number: 48

Appeared in Year: 2007

### Describe in Detail

Let C denote the Boundary of the square whose sides are given by the lines . Assume that C is described in the position sense, i. e. anticlockwise. Evaluate

### Explanation

To evaluate

Take

Here is a pole of order 1 and lies within the square.

These are simple poles which lie outside the square

So they will not contribute towards the residue of

… (80 more words) …

## Question number: 49

Appeared in Year: 2014

### Describe in Detail

Let be differentiable at and let . Evaluate

### Explanation

Now

So this is form (Exponential form)

Now using the result

Or if

Then

… (260 more words) …

## Question number: 50

» Residues of Complex Function

Appeared in Year: 2009

### Describe in Detail

Let C be the contour consisting of the lines , described counterclockwise in the plane. Compute

### Explanation

is the contour shown in the figure

To evaluate

Now is a pole of order 1 (simple pole)

Also lies within C

=

By Cauchy residue theorem

… (38 more words) …

## Question number: 51

» Taylor Series Expansion (Complex)

Appeared in Year: 2017

### Describe in Detail

Let Write down the Taylor Expansion of f in the neighbourhood of .

### Explanation

Here

Put

… (22 more words) …